
// Make newform 1098.2.a.k in Magma, downloaded from the LMFDB on 22 February 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1098_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1098_2_a_k();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1098_2_a_k();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1098_a();"
function MakeCharacter_1098_a()
    N := 1098;
    order := 1;
    char_gens := [245, 307];
    v := [1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_1098_a_Hecke(Kf)
    return MakeCharacter_1098_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[1], [0], [2], [4], [4], [-2], [-6], [4], [-8], [-10], [4], [6], [-2], [-8], [8], [6], [-12], [1], [0], [0], [10], [-12], [12], [-6], [2], [-2], [-16], [12], [14], [6], [-16], [-4], [-2], [-8], [10], [20], [14], [20], [0], [-18], [-20], [-2], [0], [2], [18], [-24], [-16], [-4], [4], [-10], [10], [8], [2], [-12], [6], [24], [-22], [24], [-2], [18], [4], [-22], [8], [0], [10], [18], [8], [-6], [-4], [-34], [-18], [-8], [-8], [-10], [28], [16], [30], [-18], [-6], [34], [4], [14], [0], [2], [24], [12], [-10], [18], [18], [-32], [-4], [40], [-16], [-12], [-16], [0], [-34], [10], [16], [-42], [-8], [22], [-28], [14], [-12], [18], [12], [-22], [24], [-22], [-48], [38], [42], [-12], [12], [2], [-16], [8], [30], [-44], [-42], [34], [14], [44], [28], [46], [-2], [0], [32], [14], [-40], [24], [8], [22], [-22], [50], [26], [16], [26], [-50], [-40], [-10], [-4], [-12], [30], [-40], [22], [-18], [-44], [-24], [-34], [30], [-48], [40], [-24], [56], [32], [6], [-38], [-18], [-36], [18], [32], [-28], [-34], [-16], [-8], [-26], [26], [-2], [12], [-2], [-32], [10], [36], [6], [-60], [-10], [-52], [14], [-4], [-60], [-26], [30], [16], [6], [14], [-44], [18], [32], [-6], [-12], [-36], [10], [-60], [-10], [-62], [-26], [54], [-16], [42], [-28], [-18], [10], [12], [-6], [-44], [60], [50], [-40], [-14], [42], [16], [36], [-32], [58], [56], [-50], [32], [22], [22], [64], [18], [56], [20], [54], [-30], [24], [44], [-60], [6], [-52], [44], [-30], [20], [8], [-30], [-2], [20], [24], [12], [-24], [-52], [46], [42], [40], [-16], [28], [40], [24], [6], [-2], [-16], [66], [-30], [-20], [62], [68], [-18], [74], [80], [60], [70], [-66], [-78], [-28], [66], [14], [68], [-22], [-34], [-4], [58], [20], [-22], [32], [-4], [-2], [26], [44], [-56], [24], [64], [22], [72], [24], [-6], [-78], [-8], [-22], [-42], [4], [6], [-36], [14], [-14], [-16], [-34], [-76], [8], [-54], [26], [-56], [-28], [-32], [34], [-76], [14], [64], [38], [-72], [50], [10], [-4], [88], [10], [-44], [-56], [-46], [-6], [-28], [-38], [38], [-36], [50], [-86], [-24], [56], [32], [6], [30], [-38], [4], [-4], [-64], [-36], [-34], [30], [50], [60], [-74], [-94], [-30], [36], [42], [36], [14], [-24], [-48], [-54], [-16], [2], [-34], [-32], [70], [-58], [-64], [20], [-78], [0], [-26], [6], [-24], [20], [4], [-6], [46], [-4], [74], [68], [-80], [56], [34], [-28], [78], [-92], [-40], [-14], [22], [2], [-58], [34], [-28], [62], [52], [24], [-72], [-50], [84], [0], [34], [34], [28], [-48], [-48], [18], [-20], [70], [52], [42], [-34], [-94], [32], [-6], [-66], [-8], [-82], [14], [4], [60], [18], [54], [-60], [76], [-62], [70], [-96], [-72], [58], [-24], [-102], [-26], [-32], [-36], [-38], [14], [-60], [-106], [0], [64]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1098_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1098_2_a_k();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_1098_2_a_k(:prec:=1)
    chi := MakeCharacter_1098_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1098_2_a_k();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_1098_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1098_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<7,R![-4, 1]>,<11,R![-4, 1]>],Snew);
    return Vf;
end function;